Comments 0

Document transcript

Chapter 5Motion of a charged particlein a magnetic ﬁeldHitherto,we have focussed on applications of quantummechanics to free parti-cles or particles conﬁned by scalar potentials.In the following,we will addressthe inﬂuence of a magnetic ﬁeld on a charged particle.Classically,the force ona charged particle in an electric and magnetic ﬁeld is speciﬁed by theLorentzforce law:Hendrik Antoon Lorentz 1853-1928A Dutch physi-cist who sharedthe 1902 NobelPrize in Physicswith Pieter Zee-man for the dis-covery and the-oretical explana-tion of the Zee-man e!ect.Healso derived the transformation equa-tions subsequently used by AlbertEinstein to describe space and time.F=q(E+v!B),whereqdenotes the charge andvthe velocity.(Here we will adopt a conventionin whichqdenotes the charge (which may be positive or negative) ande"|e|denotes themodulusof the electron charge,i.e.for an electron,the chargeq=#e=#1.602176487!10!19C.) The velocity-dependent force associatedwith the magnetic ﬁeld is quite di!erent fromthe conservative forces associatedwith scalar potentials,and the programme for transferring from classical toquantum mechanics - replacing momenta with the appropriate operators - hasto be carried out with more care.As preparation,it is helpful to revise howthe Lorentz force arises in the Lagrangian formulation of classical mechanics.Joseph-Louis Lagrange,bornGiuseppe Lodovico Lagrangia1736-1813An Italian-bornmathematicianand astronomer,who lived mostof his life inPrussia andFrance,mak-ing signiﬁcantcontributionsto all ﬁelds of analysis,to numbertheory,and to classical and celestialmechanics.On the recommendationof Euler and D’Alembert,in 1766Lagrange succeeded Euler as thedirector of mathematics at thePrussian Academy of Sciences inBerlin,where he stayed for overtwenty years,producing a largebody of work and winning severalprizes of the French Academy ofSciences.Lagrange’s treatise onanalytical mechanics,written inBerlin and ﬁrst published in 1788,o!ered the most comprehensivetreatment of classical mechanicssince Newton and formed a basis forthe development of mathematicalphysics in the nineteenth century.5.1 Classical mechanics of a particle in a ﬁeldFor a systemwithmdegrees of freedomspeciﬁed by coordinatesq1,∙ ∙ ∙qm,theclassical action is determined from the LagrangianL(qi,˙qi) byS[qi] =!dt L(qi,˙qi).The action is said to be afunctionalof the coordinatesqi(t).AccordingtoHamilton’s extremal principle(also known as theprinciple of leastaction),the dynamics of a classical system is described by the equations thatminimize the action.These equations of motion can be expressed through theclassical Lagrangian in the form of the Euler-Lagrange equations,ddt(!˙qiL(qi,˙qi))#!qiL(qi,˙qi) = 0.(5.1)"Info.Euler-Lagrange equations:According to Hamilton’s extremal princi-ple,for any smooth set of curveswi(t),the variation of the action around the classicalsolutionqi(t) is zero,i.e.lim!!01!(S[qi+#wi]#S[qi]) = 0.Applied to the action,Advanced Quantum Physics5.1.CLASSICAL MECHANICS OF A PARTICLE IN A FIELD 45the variation implies that,for anyi,"dt(wi!qiL(qi,˙qi) + ˙wi!˙qiL(qi,˙qi)) = 0.Then,integrating the second term by parts,and droping the boundary term,one obtains!dt wi#!qiL(qi,˙qi)#ddt!˙qiL(qi,˙qi)$= 0.Since this equality must follow for any functionwi(t),the term in parentheses in theintegrand must vanish leading to the Euler-Lagrange equation (5.1).Thecanonical momentumis speciﬁed by the equationpi=!˙qiL,andthe classical Hamiltonian is deﬁned by the Legendre transform,H(qi,pi) =%ipiqi#L(qi,˙qi).(5.2)It is straightforward to check that the equations of motion can be written inthe form of Hamilton’s equations of motion,˙qi=!piH,˙pi=#!qiH.From these equations it follows that,if the Hamiltonian is independent of aparticular coordinateqi,the corresponding momentumpiremains constant.Forconservative forces,1the classical Lagrangian and Hamiltonian can bewritten asL=T#V,H=T+V,withTthe kinetic energy andVthepotential energy.Sim´eon Denis Poisson 1781-1840A Frenchmathematician,geometer,andphysicist whosemathematicalskills enabledhim to computethe distributionof electricalcharges on the surface of conduc-tors.He extended the work of hismentors,Pierre Simon Laplace andJoseph Louis Lagrange,in celestialmechanics by taking their results to ahigher order of accuracy.He was alsoknown for his work in probability."Info.Poisson brackets:Any dynamical variablefin the system is somefunction of the phase space coordinates,theqis andpis,and (assuming it does notdepend explicitly on time) its time-development is given by:ddtf(qi,pi) =!qif˙qi+!pif˙pi=!qif!piH#!pif!qiH"{f,H}.The curly brackets are known as Poisson brackets,and are deﬁned for any dynamicalvariables as{A,B}=!qiA!piB#!piA!qiB.From Hamilton’s equations,we haveshown that for any variable,˙f={f,H}.It is easy to check that,for the coordinatesand canonical momenta,{qi,qj}= 0 ={pi,pj},{qi,pj}=$ij.This was theclassical mathematical structure that led Dirac to link up classical and quantummechanics:He realized that the Poisson brackets were the classical version of thecommutators,so a classical canonical momentum must correspond to the quantumdi!erential operator in the corresponding coordinate.2With these foundations revised,we now return to the problem at hand;theinﬂeunce of an electromagnetic ﬁeld on the dynamics of the charged particle.As the Lorentz force is velocity dependent,it can not be expressed simplyas the gradient of some potential.Nevertheless,the classical path traversed bya charged particle is still specifed by the principle of least action.The electricand magnetic ﬁelds can be written in terms of a scalar and a vector potentialasB=$!A,E=#$%#˙A.The corresponding Lagrangian takes the form:3L=12mv2#q%+qv∙A.1i.e.forces that conserve mechanical energy.2For a detailed discussion,we refer to Paul A.M.Dirac,Lectures on Quantum Mechanics,Belfer Graduate School of Science Monographs Series Number 2,1964.3In a relativistic formulation,the interaction termhere looks less arbitrary:the relativisticversion would have the relativistically invariantqRAµdxµadded to the action integral,wherethe four-potentialAµ= (!,A) anddxµ= (ct,dx1,dx2,dx3).This is the simplest possibleinvariant interaction between the electromagnetic ﬁeld and the particle’s four-velocity.Then,in the non-relativistic limit,qRAµdxµjust becomesqR(v∙A!!)dt.Advanced Quantum Physics5.2.QUANTUM MECHANICS OF A PARTICLE IN A FIELD 46In this case,the general coordinatesqi"xi= (x1,x2,x3) are just the Carte-sian coordinates specifying the position of the particle,and the ˙qiare the threecomponents ˙xi= ( ˙x1,˙x2,˙x3) of the particle velocities.The important point isthat thecanonicalmomentumpi=!˙xiL=mvi+qAi,is no longer simply given by the mass!velocity – there is an extra term!Making use of the deﬁnition (5.2),the corresponding Hamiltonian is givenbyH(qi,pi) =%i(mvi+qAi)vi#12mv2+q%#qv∙A=12mv2+q%.Reassuringly,the Hamiltonian just has the familiar form of the sum of thekinetic and potential energy.However,to get Hamilton’s equations of motion,the Hamiltonian has to be expressed solely in terms of the coordinates andcanonical momenta,i.e.H=12m(p#qA(r,t))2+q%(r,t).Let us now consider Hamilton’s equations of motion,˙xi=!piHand˙pi=#!xiH.The ﬁrst equation recovers the expression for the canonicalmomentum while second equation yields the Lorentz force law.To under-stand how,we must ﬁrst keep in mind thatdp/dtis not the acceleration:TheA-dependent term also varies in time,and in a quite complicated way,sinceit is the ﬁeld at a point moving with the particle.More precisely,˙pi=m¨xi+q˙Ai=m¨xi+q&!tAi+vj!xjAi',where we have assumed a summation over repeated indicies.The right-handside of the second of Hamilton’s equation,˙pi=#!H!xi,is given by#!xiH=1m(p#qA(r,t))q!xiA#q!xi%(r,t) =qvj!xiAj#q!xi%.Together,we obtain the equation of motion,m¨xi=#q&!tAi+vj!xjAi'+qvj!xiAj#q!xi%.Using the identity,v!($!A) =$(v∙A)#(v∙ $)A,andthe expressions for the electric and magnetic ﬁelds in terms of the potentials,one recovers the Lorentz equationsm¨x=F=q(E+v!B).With these preliminary discussions of the classical systemin place,we are nowin a position to turn to the quantum mechanics.5.2 Quantum mechanics of a particle in a ﬁeldTo transfer to the quantummechanical regime,we must once again implementthe canonical quantization procedure settingˆp=#i!$,so that [ˆxi,ˆpj] =i!$ij.However,in this case,ˆpi%=mˆvi.This leads to the novel situation thatthe velocities in di!erent directions do not commute.4To explore inﬂuence ofthe magnetic ﬁeld on the particle dynamics,it is helpful to assess the relativeweight of theA-dependent contributions to the quantum Hamiltonian,ˆH=12m(ˆp#qA(r,t))2+q%(r,t).4Withmˆvi=!i!"xi!qAi,it is easy (and instructive) to verify that [ˆvx,ˆvy] =i!qm2B.Advanced Quantum Physics5.3.ATOMIC HYDROGEN:NORMAL ZEEMAN EFFECT 47Expanding the square on the right hand side of the Hamiltonian,thecross-term (known as theparamagnetic term) leads to the contribution#q!2im($∙A+A∙ $) =iq!mA∙ $,where the last equality follows from theCoulomb gauge condition,$∙A= 0.5Combined with thediamagnetic(A2)contribution,one obtains the Hamiltonian,ˆH=#!22m$2+iq!mA∙ $+q22mA2+q%.For a constant magnetic ﬁeld,the vector potential can be written asA=#r!B/2.In this case,the paramagnetic component takes the formiq!mA∙ $=iq!2m(r!$)∙B=#q2mL∙B,whereLdenotes the angular momentum operator (with the hat not shown forbrevity!).Similarly,the diamagnetic term leads toq22mA2=q28m&r2B2#(r∙B)2'=q2B28m(x2+y2),where,here,we have chosen the magnetic ﬁeld to lie along thez-axis.5.3 Atomic hydrogen:Normal Zeeman e!ectBefore addressing the role of these separate contributions in atomic hydrogen,let us ﬁrst estimate their relative magnitude.With&x2+y2'(a20,wherea0denotes the Bohr radius,and&Lz'(!,the ratio of the paramagnetic anddiamagnetic terms is given by(q2/8me)&x2+y2'B2(q/2me)&Lz'B=e4a20B2!B(10!6B/T.Therefore,while electrons remain bound to atoms,for ﬁelds that can beachieved in the laboratory (B(1 T),the diamagnetic term is negligible ascompared to the paramagnetic term.Moreover,when compared with theCoulomb energy scale,eB!/2memec2&2/2=e!(mec&)2B(B/T2.3!105,where&=e24"#01!c(1137denotes the ﬁne structure constant,one may seethat the paramagnetic term provides only a small perturbation to the typicalatomic splittings.Splitting of the sodium D linesdue to an external magnetic ﬁeld.The multiplicity of the lines andtheir “selection rule” will be dis-cussed more fully in chapter 9.The ﬁgure is taken fromthe orig-inal paper,P.Zeeman,The e!ectof magnetization on the nature oflight emitted by a substance,Na-ture55,347 (1897).5The electric ﬁeldEand magnetic ﬁeldBof Maxwell’s equations contain only “physical”degrees of freedom,in the sense that every mathematical degree of freedomin an electromag-netic ﬁeld conﬁguration has a separately measurable e!ect on the motions of test charges inthe vicinity.As we have seen,these “ﬁeld strength” variables can be expressed in terms ofthe scalar potential!and the vector potentialAthrough the relations:E=!"!!"tAandB="#A.Notice that ifAis transformed toA+"",Bremains unchanged,sinceB="#[A+""] ="#A.However,this transformation changesEasE=!"!!"tA!""t"=!"[!+"t"]!"tA.If!is further changed to!!"t",Eremains unchanged.Hence,both theEandBﬁeldsare unchanged if we take any function"(r,t) and simultaneously transformA$A+""!$!!"t".A particular choice of the scalar and vector potentials is agauge,and a scalar function"used to change the gauge is called a gauge function.The existence of arbitrary numbers ofgauge functions"(r,t),corresponds to the U(1) gauge freedom of the theory.Gauge ﬁxingcan be done in many ways.Advanced Quantum Physics5.4.GAUGE INVARIANCE AND THE AHARONOV-BOHM EFFECT 48However,there are instances when the diamagnetic contriubution can playan important role.Leaving aside the situation that may prevail on neutronstars,where magnetic ﬁelds as high as 108T may exist,the diamagnetic con-tribution can be large when the typical “orbital” scale&x2+y2'becomesmacroscopic in extent.Such a situation arises when the electrons becomeunbound such as,for example,in a metal or a synchrotron.For a furtherdiscussion,see section 5.5 below.Retaining only the paramagnetic contribution,the Hamiltonian for a “spin-less” electron moving in a Coulomb potential in the presence of a constantmagnetic ﬁeld then takes the form,ˆH=ˆH0+e2mBLz,whereˆH0=ˆp22m#e24"#0r.Since [ˆH0,Lz] = 0,the eigenstates of the unperturbedHamiltonian,'l$m(r) remain eigenstates ofˆHand the corresponding energylevels are speciﬁed byEn$m=#Ryn2+!(Lmwhere(L=eB2mdenotes theLarmor frequency.From this result,we expectthat a constant magnetic ﬁeld will lead to a splitting of the (2)+1)-fold degen-eracy of the energy levels leading to multiplets separated by a constant energyshift of!(L.The fact that this behaviour is not recapitulated generically byexperiment was one of the key insights that led to the identiﬁcation of electronspin,a matter to which we will turn in chapter 6.Sir Joseph Larmor 1857-1942A physicist andmathematicianwho made in-novations in theunderstandingof electricity,dynamics,ther-modynamics,and the electron theory of matter.His most inﬂuential work wasAetherand Matter,a theoretical physicsbook published in 1900.In 1903 hewas appointed Lucasian Professor ofMathematics at Cambridge,a posthe retained until his retirement in1932.5.4 Gauge invariance and the Aharonov-Bohm ef-fectOur derivation above shows that the quantum mechanical Hamiltonian of acharged particle is deﬁned in terms of the vector potential,A.Since the latteris deﬁned only up to some gauge choice,this suggests that the wavefunctionis not a gauge invariant object.Indeed,it is only the observables associatedwith the wavefunction which must be gauge invariant.To explore this gaugefreedom,let us consider the inﬂuence of thegauge transformation,A)*A"=A+$",%)*%"#!t",where"(r,t) denotes a scalar function.Under the gauge transformation,onemay show that the corresponding wavefunction gets transformed as'"(r,t) = exp(iq!"(r,t))'(r,t).(5.3)"Exercise.If wavefunction'(r,t) obeys the time-dependent Schr¨odinger equa-tion,i!!t'=ˆH[A,%]',show that'"(r,t) as deﬁned by (5.3) obeys the equationi!!t'"=ˆH"[A",%"]'".The gauge transformation introduces an additional space and time-dependentphase factor into the wavefunction.However,since the observable translatesto the probability density,|'|2,this phase dependence seems invisible."Info.One physical manifestation of the gauge invariance of the wavefunctionis found in theAharonov-Bohme!ect.Consider a particle with chargeqtravellingAdvanced Quantum Physics5.4.GAUGE INVARIANCE AND THE AHARONOV-BOHM EFFECT 49Figure 5.1:(Left) Schematic showing the geometry of an experiment to observe theAharonov-Bohm e!ect.Electrons from a coherent source can follow two paths whichencircle a region where the magnetic ﬁeld is non-zero.(Right) Interference fringesfor electron beams passing near a toroidal magnet from the experiment by Tonomuraand collaborators in 1986.The electron beam passing through the center of the torusacquires an additional phase,resulting in fringes that are shifted with respect tothose outside the torus,demonstrating the Aharonov-Bohm e!ect.For details see theoriginal paper from which this image was borrowed see Tonomuraet al.,Evidencefor Aharonov-Bohm e!ect with magnetic ﬁeld completely shielded from electron wave,Phys.Rev.Lett.56,792 (1986).along a path,P,in which the magnetic ﬁeld,B= 0 is identically zero.However,avanishing of the magnetic ﬁeld does not imply that the vector potential,Ais zero.Indeed,as we have seen,any"(r) such thatA=$"will translate to this condition.In traversing the path,the wavefunction of the particle will acquire the phase factor*=q!"PA∙dr,where the line integral runs along the path.If we consider now two separate pathsPandP"which share the same initial andﬁnal points,the relative phase of the wavefunction will be set by#*=q!!PA∙dr#q!!P!A∙dr=q!*A∙dr=q!!AB∙d2r,where the line integral+runs over the loop involving pathsPandP",and"Arunsover the area enclosed by the loop.The last relation follows from the application ofStokes’ theorem.This result shows that the relative phase#*is ﬁxed by the factorq/!multiplied by the magnetic ﬂux $ ="AB∙d2renclosed by the loop.6In theabsence of a magnetic ﬁeld,the ﬂux vanishes,and there is no additional phase.Sir George Gabriel Stokes,1stBaronet 1819-1903A mathematicianand physicist,who at Cam-bridge madeimportant con-tributions toﬂuid dynamics(including theNavierStokesequations),optics,and mathematicalphysics (including Stokes’ theorem).He was secretary,and then president,of the Royal Society.However,if we allow the paths to enclose a region of non-vanishing magneticﬁeld (see ﬁgure 5.1(left)),even if the ﬁeld is identically zero on the pathsPandP",the wavefunction will acquire a non-vanishing relative phase.This ﬂux-dependentphase di!erence translates to an observable shift of interference fringes when on anobservation plane.Since the original proposal,7the Aharonov-Bohm e!ect has beenstudied in several experimental contexts.Of these,the most rigorous study was un-dertaken by Tonomura in 1986.Tomomura fabricated a doughnut-shaped (toroidal)ferromagnet six micrometers in diameter (see ﬁgure 5.1b),and covered it with a nio-biumsuperconductor to completely conﬁne the magnetic ﬁeld within the doughnut,inaccordance with the Meissner e!ect.8With the magnet maintained at 5K,they mea-sured the phase di!erence from the interference fringes between one electron beampassing though the hole in the doughnut and the other passing on the outside ofthe doughnut.The results are shown in ﬁgure 5.1(right,a).Interference fringes aredisplaced with just half a fringe of spacing inside and outside of the doughnut,indi-cating the existence of the Aharonov-Bohm e!ect.Although electrons pass throughregions free of any electromagnetic ﬁeld,an observable e!ect was produced due to theexistence of vector potentials.6Note that the phase di!erence depends on the magnetic ﬂux,a function of the magneticﬁeld,and is therefore a gauge invariant quantity.7Y.Aharonov and D.Bohm,Signiﬁcance of electromagnetic potentials in quantum theory,Phys.Rev.115,485 (1959).8Perfect diamagnetism,a hallmark of superconductivity,leads to the complete expulsionof magnetic ﬁelds – a phenomenon known as the Meissner e!ect.Advanced Quantum Physics5.5.FREE ELECTRONS IN A MAGNETIC FIELD:LANDAU LEVELS 50The observation of the half-fringe spacing reﬂects the constraints imposed bythe superconducting toroidal shield.When a superconductor completely surroundsa magnetic ﬂux,the ﬂux is quantized to an integral multiple of quantized ﬂuxh/2e,the factor of two reﬂecting that fact that the superconductor involves a condensate ofelectronpairs.When an odd number of vortices are enclosed inside the superconduc-tor,the relative phase shift becomes+(mod.2+) – half-spacing!For an even numberof vortices,the phase shift is zero.95.5 Free electrons in a magnetic ﬁeld:Landau levelsFinally,to complete our survey of the inﬂuence of a uniform magnetic ﬁeld onthe dynamics of charged particles,let us consider the problem of a free quan-tum particle.In this case,the classical electron orbits can be macroscopic andthere is no reason to neglect the diamagnetic contribution to the Hamiltonian.Previously,we have worked with a gauge in whichA= (#y,x,0)B/2,giving aconstant ﬁeldBin thez-direction.However,to address the Schr¨odinger equa-tion for a particle in a uniform perpendicular magnetic ﬁeld,it is convenientto adopt theLandau gauge,A(r) = (#By,0,0).Lev Davidovich Landau 1908-1968A prominentSoviet physicistwho madefundamentalcontributionsto many areasof theoreticalphysics.Hisaccomplishmentsinclude theco-discovery of the density matrixmethod in quantum mechanics,the quantum mechanical theory ofdiamagnetism,the theory of super-ﬂuidity,the theory of second orderphase transitions,the Ginzburg-Landau theory of superconductivity,the explanation of Landau dampingin plasma physics,the Landau polein quantum electrodynamics,and thetwo-component theory of neutrinos.He received the 1962 Nobel Prizein Physics for his development of amathematical theory of superﬂuiditythat accounts for the properties ofliquid helium II at a temperaturebelow 2.17K."Exercise.Construct the gauge transformation,"(r) which connects thesetwo representations of the vector potential.In this case,the stationary form of the Schr¨odinger equation is given byˆH'(r) =12m,(ˆpx+qBy)2+ ˆp2y+ ˆp2z-'(r) =E'(r).SinceˆHcommutes with both ˆpxand ˆpz,both operators have a common set ofeigenstates reﬂecting the fact thatpxandpzare conserved by the dynamics.The wavefunction must therefore take the form,'(r) =ei(pxx+ipzz)/!,(y),with,(y) deﬁned by the equation,.ˆpy22m+12m(2(y#y0)2/,(y) =#E#p2z2m$,(y).Herey0=#px/qBand(=|q|B/mcoincides with thecyclotron frequencyof the classical charged particle (exercise).We now see that the conservedcanonical momentumpxin thex-direction is in fact the coordinate of the centreof a simple harmonic oscillator potential in they-direction with frequency(.As a result,we can immediately infer that the eigenvalues of the Hamiltonianare comprised of a free particle component associated with motion parallel tothe ﬁeld,and a set of harmonic oscillator states,En,pz= (n+1/2)!(+p2z2m.The quantum numbers,n,specify states known asLandau levels.Let us conﬁne our attention to states corresponding to the lowest oscillator(Landau level) state,(and,for simplicity,pz= 0),E0=!(/2.What isthe degeneracy of this Landau level?Consider a rectangular geometry ofareaA=Lx!Lyand,for simplicity,take the boundary conditions to beperiodic.The centre of the oscillator wavefunction,y0=#px/qB,must lie9The superconducting ﬂux quantumwas actually predicted prior to Aharonov and Bohm,by Fritz London in 1948 using a phenomenological theory.Advanced Quantum Physics5.5.FREE ELECTRONS IN A MAGNETIC FIELD:LANDAU LEVELS 51between 0 andLy.With periodic boundary conditionseipxLx/!= 1,so thatpx=n2+!/Lx.This means thaty0takes a series of evenly-spaced discretevalues,separated by#y0=h/qBLx.So,for electron degrees of freedom,q=#e,the total number of statesN=Ly/|#y0|,i.e.-max=LxLyh/eB=AB$0,(5.4)where $0=e/hdenotes the “ﬂux quantum”.So the total number of states inthe lowest energy level coincides with the total number of ﬂux quanta makingup the ﬁeldBpenetrating the areaA.Klaus von Klitzing,1943-German physicistwho was awardedthe Nobel Prizefor Physics in1985 for hisdiscovery thatunder appropri-ate conditionsthe resistanceo!ered by anelectrical conductor is quantized.The work was ﬁrst reported in thefollowing reference,K.v.Klitzing,G.Dorda,and M.Pepper,New methodfor high-accuracy determination ofthe ﬁne-structure constant basedon quantized Hall resistance,Phys.Rev.Lett.45,494 (1980).The Landau level degeneracy,-max,depends on ﬁeld;the larger the ﬁeld,the more electrons can be ﬁt into each Landau level.In the physical system,each Landau level is spin split by the Zeeman coupling,with (5.4) applying toone spin only.Finally,although we treatedxandyin an asymmetric manner,this was merely for convenience of calculation;no physical quantity shoulddierentiate between the two due to the symmetry of the original problem."Exercise.Consider the solution of the Schr¨odinger equation when working inthe symmetric gauge withA=#r!B/2.Hint:consider the velocity commutationrelations,[vx,vy] and how these might be deployed as conjugate variables."Info.It is instructive to infery0from purely classical considerations:Writingm˙v=qv!Bin component form,we havem¨x=qBc˙y,m¨y=#qBc˙x,andm¨z= 0.Focussing on the motion in thexy-plane,these equations integrate straightforwardlyto give,m˙x=qBc(y#y0),m˙y=#qBc(x#x0).Here (x0,y0) are the coordinates ofthe centre of the classical circular motion (known as the “guiding centre”) – thevelocity vectorv= ( ˙x,˙y) always lies perpendicular to (r#r0),andr0is given byy0=y#mvx/qB=#px/qB,x0=x+mvy/qB=x+py/qB.(Recall that we are using the gaugeA(x,y,z) = (#By,0,0),andpx=!˙xL=mvx+qAx,etc.) Just asy0is a conserved quantity,so isx0:it commutes with theHamiltonian since [x+cˆpy/qB,ˆpx+qBy] = 0.However,x0andy0do not commutewith each other:[x0,y0] =#i!/qB.This is why,when we chose a gauge in whichy0was sharply deﬁned,x0was spread over the sample.If we attempt to localize thepoint (x0,y0) as much as possible,it is smeared out over an area corresponding toone ﬂux quantum.The natural length scale of the problem is therefore the magneticlength deﬁned by)=0!qB."Info.Integer quantum Hall e!ect:Until now,we have considered theimpact of just a magnetic ﬁeld.Consider now the Hall e!ect geometry in whichwe apply a crossed electric,Eand magnetic ﬁeld,B.Taking into account bothcontributions,the total current ﬂow is given byj=.0#E#j!Bne$,where.0denotes the conductivity,andnis the electron density.With the electricﬁeld oriented alongy,and the magnetic ﬁeld alongz,the latter equation may berewritten as#1"0Bne#"0Bne1$#jxjy$=.0#0Ey$.Inverting these equations,one ﬁnds thatjx=#.20B/ne1 +(.0B/ne)21234"xyEy,jy=.01 +(.0B/ne)21234"yyEy.Advanced Quantum Physics5.5.FREE ELECTRONS IN A MAGNETIC FIELD:LANDAU LEVELS 52Figure 5.2:(Left) A voltageVdrives a currentIin the positivexdirection.NormalOhmic resistance isV/I.A magnetic ﬁeld in the positivezdirection shifts positivecharge carriers in the negativeydirection.This generates a Hall potential and aHall resistance (V H/I) in theydirection.(Right) The Hall resistance varies stepwisewith changes in magnetic ﬁeldB.Step height is given by the physical constanth/e2(value approximately 25 k%) divided by an integeri.The ﬁgure shows steps fori= 2,3,4,5,6,8 and 10.The e!ect has given rise to a new international standardfor resistance.Since 1990 this has been represented by the unit 1 klitzing,deﬁned asthe Hall resistance at the fourth step (h/4e2).The lower peaked curve represents theOhmic resistance,which disappears at each step.These provide the classical expressions for the longitudinal and Hall conductivities,.yyand.xyin the crossed ﬁeld.Note that,for these classical expressions,.xyisproportional toB.How does quantum mechanics revised this picture?For the classical model –Drude theory,the random elastic scattering of electrons impurities leads to a con-stant drift velocity in the presence of a constant electric ﬁeld,.0=ne2#me,where/denotes the mean time between collisions.Now let us suppose the magnetic ﬁeld ischosen so that number of electrons exactly ﬁlls all the Landau levels up to someN,i.e.nLxLy=N-max+n=NeBh.The scattering of electrons must lead to a transfer between quantumstates.However,if all states of the same energy are ﬁlled,10elastic (energy conserving) scatteringbecomes impossible.Moreover,since the next accessible Landau level energy is adistance!(away,at low enough temperatures,inelastic scattering becomes frozenout.As a result,the scattering time vanishes at special values of the ﬁeld,i.e..yy*0and.xy*neB=Ne2h.At critical values of the ﬁeld,the Hall conductivity is quantized in units ofe2/h.Inverting the conductivity tensor,one obtains the resistivity tensor,#0xx0xy#0xy0xx$=#.xx.xy#.xy.xx$#1,where0xx=.xx.2xx+.2xy,0xx=#.xy.2xx+.2xy,So,when.xx= 0 and.xy=-e2/h,0xx= 0 and0xy=h/-e2.The quantumHall state describes dissipationless current ﬂow in which the Hall conductance.xyisquantized in units ofe2/h.Experimental measurements of these values provides thebest determination of fundamental ratioe2/h,better than 1 part in 107.10Note that electons are subject to Pauli’s exclusion principle restricting the occupancy ofeach state to unity.Advanced Quantum Physics